{"id":"https://openalex.org/W2405626424","doi":"https://doi.org/10.1145/2930889.2930927","title":"Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems","display_name":"Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems","publication_year":2016,"publication_date":"2016-07-19","ids":{"openalex":"https://openalex.org/W2405626424","doi":"https://doi.org/10.1145/2930889.2930927","mag":"2405626424"},"language":"en","primary_location":{"id":"doi:10.1145/2930889.2930927","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2930889.2930927","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","raw_type":"proceedings-article"},"type":"preprint","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1605.05889","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5107405594","display_name":"Jean\u2010Charles Faug\u00e8re","orcid":null},"institutions":[{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I39804081","display_name":"Sorbonne Universit\u00e9","ror":"https://ror.org/02en5vm52","country_code":"FR","type":"education","lineage":["https://openalex.org/I39804081"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Jean-Charles Faug\u00e8re","raw_affiliation_strings":["INRIA/Sorbonne Universit\u00e9s/Univ. Paris 6/CNRS, Equipe PolSys, Paris, France"],"affiliations":[{"raw_affiliation_string":"INRIA/Sorbonne Universit\u00e9s/Univ. Paris 6/CNRS, Equipe PolSys, Paris, France","institution_ids":["https://openalex.org/I39804081","https://openalex.org/I1326498283","https://openalex.org/I1294671590"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056341371","display_name":"Pierre-Jean Spaenlehauer","orcid":"https://orcid.org/0000-0001-7906-0505"},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I1326498283","display_name":"Institut national de recherche en informatique et en automatique","ror":"https://ror.org/02kvxyf05","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1326498283"]},{"id":"https://openalex.org/I90183372","display_name":"Universit\u00e9 de Lorraine","ror":"https://ror.org/04vfs2w97","country_code":"FR","type":"education","lineage":["https://openalex.org/I90183372"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Pierre-Jean Spaenlehauer","raw_affiliation_strings":["INRIA/CNRS/Universit\u00e9 de Lorraine, Nancy, France"],"affiliations":[{"raw_affiliation_string":"INRIA/CNRS/Universit\u00e9 de Lorraine, Nancy, France","institution_ids":["https://openalex.org/I90183372","https://openalex.org/I1326498283","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5003800445","display_name":"Jules Svartz","orcid":null},"institutions":[{"id":"https://openalex.org/I4210141076","display_name":"Ministere de l'Education Nationale","ror":"https://ror.org/04xp49e10","country_code":"FR","type":"funder","lineage":["https://openalex.org/I2802818602","https://openalex.org/I4210141076"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Jules Svartz","raw_affiliation_strings":["Lyc\u00e9e Mass\u00e9na/Min. Education Nationale, Nice, France"],"affiliations":[{"raw_affiliation_string":"Lyc\u00e9e Mass\u00e9na/Min. Education Nationale, Nice, France","institution_ids":["https://openalex.org/I4210141076"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5107405594"],"corresponding_institution_ids":["https://openalex.org/I1294671590","https://openalex.org/I1326498283","https://openalex.org/I39804081"],"apc_list":null,"apc_paid":null,"fwci":0.3596,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.66428188,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"31","issue":null,"first_page":"223","last_page":"230"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9976000189781189,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9872999787330627,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7424271702766418},{"id":"https://openalex.org/keywords/monomial","display_name":"Monomial","score":0.7389128804206848},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.5480464100837708},{"id":"https://openalex.org/keywords/system-of-polynomial-equations","display_name":"System of polynomial equations","score":0.5386921763420105},{"id":"https://openalex.org/keywords/polytope","display_name":"Polytope","score":0.5095691680908203},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.47276216745376587},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4657161235809326},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4644226133823395},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46071112155914307},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4585757851600647},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10311508178710938},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08821609616279602}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7424271702766418},{"id":"https://openalex.org/C11252640","wikidata":"https://www.wikidata.org/wiki/Q243723","display_name":"Monomial","level":2,"score":0.7389128804206848},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.5480464100837708},{"id":"https://openalex.org/C105335568","wikidata":"https://www.wikidata.org/wiki/Q7663735","display_name":"System of polynomial equations","level":3,"score":0.5386921763420105},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.5095691680908203},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.47276216745376587},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4657161235809326},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4644226133823395},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46071112155914307},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4585757851600647},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10311508178710938},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08821609616279602},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1145/2930889.2930927","is_oa":false,"landing_page_url":"https://doi.org/10.1145/2930889.2930927","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation","raw_type":"proceedings-article"},{"id":"pmh:oai:arXiv.org:1605.05889","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.05889","pdf_url":"https://arxiv.org/pdf/1605.05889","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},{"id":"pmh:oai:HAL:hal-01314651v1","is_oa":false,"landing_page_url":"https://inria.hal.science/hal-01314651","pdf_url":null,"source":{"id":"https://openalex.org/S4306402512","display_name":"HAL (Le Centre pour la Communication Scientifique Directe)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1294671590","host_organization_name":"Centre National de la Recherche Scientifique","host_organization_lineage":["https://openalex.org/I1294671590"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"International Symposium on Symbolic and Algebraic Computation (ISSAC 2016), Jul 2016, Waterloo, Canada. pp.223-230, &#x27E8;10.1145/2930889.2930927&#x27E9;","raw_type":"Conference papers"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1605.05889","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1605.05889","pdf_url":"https://arxiv.org/pdf/1605.05889","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":null},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2405626424.pdf","grobid_xml":"https://content.openalex.org/works/W2405626424.grobid-xml"},"referenced_works_count":34,"referenced_works":["https://openalex.org/W647687692","https://openalex.org/W1495410784","https://openalex.org/W1496428155","https://openalex.org/W1504054803","https://openalex.org/W1537258143","https://openalex.org/W1984254060","https://openalex.org/W1985924702","https://openalex.org/W1988709433","https://openalex.org/W1991675728","https://openalex.org/W1991813962","https://openalex.org/W2001588532","https://openalex.org/W2006394656","https://openalex.org/W2015508922","https://openalex.org/W2018980874","https://openalex.org/W2032831657","https://openalex.org/W2032925102","https://openalex.org/W2033016032","https://openalex.org/W2058786070","https://openalex.org/W2063194632","https://openalex.org/W2065276731","https://openalex.org/W2066979034","https://openalex.org/W2072992931","https://openalex.org/W2082387287","https://openalex.org/W2104223485","https://openalex.org/W2118254039","https://openalex.org/W2124706092","https://openalex.org/W2130918732","https://openalex.org/W2134386365","https://openalex.org/W2151054383","https://openalex.org/W2164838307","https://openalex.org/W2279788372","https://openalex.org/W2572028317","https://openalex.org/W2592887901","https://openalex.org/W4206272148"],"related_works":["https://openalex.org/W2399505800","https://openalex.org/W2018253224","https://openalex.org/W333749553","https://openalex.org/W1966623661","https://openalex.org/W2942288111","https://openalex.org/W2039516134","https://openalex.org/W180925886","https://openalex.org/W2075184731","https://openalex.org/W2948218175","https://openalex.org/W4365211605"],"abstract_inverted_index":{"Bezout's":[0],"theorem":[1],"states":[2],"that":[3,55,115,196,199,215,238,270],"dense":[4],"generic":[5,56],"systems":[6,57,278],"of":[7,28,38,50,76,79,92,104,135,146,155,157,164,181,209,226,257],"n":[8,13,246],"multivariate":[9],"quadratic":[10,51,125,228],"equations":[11,33],"in":[12,31,45,109,166,260,272],"variables":[14,283],"have":[15],"2n":[16],"solutions":[17,39],"over":[18],"algebraically":[19],"closed":[20],"fields.":[21],"When":[22],"only":[23],"a":[24,124,136,153,178],"small":[25],"subset":[26],"M":[27,53,60,147,217],"monomials":[29,52,229],"appear":[30],"the":[32,36,74,90,101,105,117,132,144,162,167,197,200,207,224,236,239,254,261],"(fewnomial":[34],"systems),":[35],"number":[37,134,163,180,208,256],"may":[40],"decrease":[41],"dramatically.":[42],"We":[43],"focus":[44],"this":[46,172,189,249],"work":[47],"on":[48,89,206],"subsets":[49,225],"such":[54,93],"with":[58,139,253,279],"support":[59],"do":[61],"not":[62],"admit":[63],"any":[64],"solution":[65],"at":[66,221,231],"all.":[67],"For":[68],"these":[69],"systems,":[70],"Hilbert's":[71],"Nullstellensatz":[72],"ensures":[73],"existence":[75],"algebraic":[77],"certificates":[78,94,271],"inconsistency.":[80],"However,":[81],"up":[82],"to":[83,243],"our":[84],"knowledge":[85],"all":[86],"known":[87],"bounds":[88],"sizes":[91],"---including":[95],"those":[96],"which":[97],"take":[98],"into":[99],"account":[100],"Newton":[102],"polytopes":[103],"polynomials---":[106],"are":[107],"exponential":[108],"n.":[110],"Our":[111],"main":[112],"results":[113,268],"show":[114,214],"if":[116,216],"inequality":[118,190,201,240],"2|M|-2n":[119],"\u2264":[120],"\u221a{1+8\u03bd}-1":[121],"holds":[122,241],"for":[123,277],"fewnomial":[126],"system":[127],"--":[128,148],"where":[129],"\u03bd":[130],"is":[131,143,202,218,251],"matching":[133,255],"graph":[137],"associated":[138],"M,":[140],"and":[141,192,284],"|M|":[142],"cardinality":[145],"then":[149,235],"there":[150],"exists":[151],"generically":[152],"certificate":[154,173],"inconsistency":[156,273],"linear":[158],"size":[159],"(measured":[160],"as":[161,245],"coefficients":[165],"ground":[168],"field":[169],"K).":[170],"Moreover":[171],"can":[174,274],"be":[175,275],"computed":[176,276],"within":[177],"polynomial":[179],"arithmetic":[182],"operations.":[183],"Next,":[184],"we":[185,193,213,265],"evaluate":[186],"how":[187],"often":[188],"holds,":[191],"give":[194],"evidence":[195],"probability":[198,237],"satisfied":[203],"depends":[204],"strongly":[205],"squares.":[210],"More":[211],"precisely,":[212],"picked":[219],"uniformly":[220],"random":[222,258],"among":[223],"n+k+1":[227],"containing":[230],"least":[232],"\u03a9(n1/2+\u03b5)":[233],"squares,":[234],"tends":[242],"1":[244],"grows.":[247],"Interestingly,":[248],"phenomenon":[250],"related":[252],"graphs":[259],"Erdos-Renyi":[262],"model.":[263],"Finally,":[264],"provide":[266],"experimental":[267],"showing":[269],"more":[280],"than":[281],"10000":[282],"equations.":[285]},"counts_by_year":[{"year":2018,"cited_by_count":1}],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}
